# Valuation of Cash-Or-Nothing option

Studying options pricing, I'm stuck with the following problem:

The price of a stock is described by the dynamic: $$dS_t = \mu\, dt + \sigma\,dW_t$$ Compute the fair price of a Cash or Nothing Option with pay-off function $$V(S_T) = \mathbb{1}_{S_T.

Hint: Replace $$\mu$$ such that, the discounted price at maturity $$S(T)$$ under the risk-free measure is a martingale.

It means that the option can just be exercised at maturity $$T$$ and has value $$1$$ when at maturity the underlying price is below the strike.

My thoughts: Use a discretization process like Euler-Maruyama and then compute recursively the value of $$S(T)$$. Then using the pay-off function, approximate it with a Monte-Carlo simulation.

However, I don't know how to use this Hint. My professor said it could be really useful but I do not know how to use it. Any help with this problem would be really meaningful.

Many thanks.

You can use such an approximation but there are known analytical prices. You have a special case in which the stock price is normally distributed. See Bachelier Model.

Set $$\mu=r-q$$ (if you have dividends, or simply $$\mu=r$$ if there are no dividends). So if you change from the real worl probability measure $$\mathbb{P}$$ to the risk-neutral measure $$\mathbb{Q}$$ you get that $$\mathrm{d}S_t=(r-q)\mathrm{d}t+\sigma \mathrm{d}W_t$$. Then, using risk-neutral pricing, the inital value of your claim is given by \begin{align*} V_0 &= e^{-rT} \mathbb{E}^\mathbb{Q}[{1}_{\{S_T< K\}}] \\ &= e^{-rT} \mathbb{Q}[\{S_T< K\}]. \end{align*}

Thus, all you need to do is to find the probability distribution of $$S_T$$ under $$\mathbb{Q}$$. Using again that $$\mathrm{d}S_t=(r-q)\mathrm{d}t+\sigma \mathrm{d}W_t$$, we see that $$(S_t)$$ is an arithmetric Brownian motion under $$\mathbb{Q}$$ and thus normally distributed. Furthermore, \begin{align*} S_T= S_0+(r-q)T + \sigma W_T \sim N\big(S_0+(r-q)T,\sigma^2T\big), \end{align*} since $$W_T\sim N(0,T)$$. Now, set $$m=S_0+(r-q)T$$ and $$s=\sigma\sqrt{T}$$. Then, $$S_T=m+sZ$$ where $$Z\sim N(0,1)$$. Thus, \begin{align*} V_0 &= e^{-rT} \mathbb{Q}[\{m+sZ< K\}]\\ &= e^{-rT} \mathbb{Q}\left[\left\{Z< \frac{K-m}{s}\right\}\right]\\ &= e^{-rT} \Phi\left(\frac{K-m}{s}\right)\\ &= e^{-rT} \Phi\left(-\frac{S_0-K+(r-q)T}{\sigma\sqrt{T}}\right) \\ &= e^{-rT} \left(1- \Phi\left(\frac{S_0-K+(r-q)T}{\sigma\sqrt{T}}\right)\right), \end{align*}

where $$\Phi$$ denotes the cumulative distribution function of a standard normal distribution.

Let me highlight that, of course, you can price such a claim with Euler Maruyama. You can also employ finite differences or Fourier transforms. You could even build a (binomial) tree. But if there is a simple analytical answer available, it is to be preferred.

By the way, in the Black-Scholes model, the price of a Cash-Or-Nothing option is given by $$e^{-rT}\Phi(-d_2)=e^{-rT}\big(1-\Phi(d_2)\big)$$, see here.